Hilbert space embeddings of conditional distributions with applications to dynamical systems

Abstract

In this paper, we extend the Hilbert space embedding approach to handle conditional distributions. We derive a kernel estimate for the conditional embedding, and show its connection to ordinary embeddings. Condi-tional embeddings largely extend our ability to manipulate distributions in Hilbert spaces, and as an example, we derive a nonpara-metric method for modeling dynamical sys-tems where the belief state of the system is maintained as a conditional embedding. Our method is very general in terms of both the domains and the types of distributions that it can handle, and we demonstrate the ef-fectiveness of our method in various dynami-cal systems. We expect that conditional em-beddings will have wider applications beyond modeling dynamical systems. 1.